Proving the Identity: $x - \frac{x^3}{3}$ for $x \in {0, 1}$
In this article, we will prove the identity $x - \frac{x^3}{3}$ for $x \in {0, 1}$. This identity may seem simple, but it has important implications in various mathematical contexts.
Case 1: $x = 0$
Let's start by plugging in $x = 0$ into the expression:
$x - \frac{x^3}{3} = 0 - \frac{0^3}{3} = 0$
As we can see, the expression evaluates to $0$ when $x = 0$, which is a true statement.
Case 2: $x = 1$
Now, let's plug in $x = 1$ into the expression:
$x - \frac{x^3}{3} = 1 - \frac{1^3}{3} = 1 - \frac{1}{3} = \frac{2}{3}$
Again, the expression evaluates to a true statement, which is $\frac{2}{3}$.
Conclusion
We have shown that the identity $x - \frac{x^3}{3}$ holds true for both $x = 0$ and $x = 1$. This means that the identity is valid for all values of $x$ in the set ${0, 1}$. This result has important implications in various mathematical contexts, such as algebra, calculus, and number theory.
Importance of the Identity
The identity $x - \frac{x^3}{3}$ has several important applications in mathematics. For example, it is used in the study of polynomial equations, where it helps to simplify complex expressions. It also has implications in calculus, where it is used to study the behavior of functions near their critical points.
In conclusion, we have proven that the identity $x - \frac{x^3}{3}$ holds true for $x \in {0, 1}$. This result has important implications in various mathematical contexts and is a fundamental tool for mathematicians and scientists.
